312 PROF. SYLVESTER AND MR. HAMMOND OX HAMILTON’S NUMBERS. 
Note 1, page 299.—(September 17, 1887.) 
It is easy to see that, if SM and Sa are corresponding errors in the values of M and 
a respectively, 
SM = (M log, M log, 2) 8a = (-38822 . . . ) Sa 
(since M = 1-46544 . . . , log, M = -38220 . . . , and log, 2 = -69314 . . . ). 
Hence, Sa being intermediate between "0000003 and "0000006, 
SM lies between "000000116 and "000000233. 
The value of M (the base of the Hamiltonian Numbers) is thus found to be 
1 "465443 . . . , correct to the last figure inclusive.—J. J. S. 
Note 2, page 300.—(September 17, 1887.) 
This equation may be obtained more simply from the fundamental formula of 
Hamilton (middle of above note). It follows from the law of derivation there given 
that, if we write 1 F„ = (1 — x)~ l F„ — x n , and, in general, j+l Y n = (1 — x)- l ->¥ n — x n , 
then F* +1 = an F, t ; and, consequently, 
F n + l — (1 — x)~ an Y n = — x' l {l -J- (1 — x)~ l + (1 — x)~ 2 + . . . + (1 — •r) _ ' I,, + 1 } 
= x n ~ l {(l — x) — [l — x)~ an+1 }. —J. J. S. 
Note 3, page 303. — (September 19, 1887.) 
It is curious to notice the sort of affinity which exists between a form of writing 
the scale of relation for Bernoulli’s Numbers and that given at p. 289 for Hamilton’s. 
If we write G 0 = 1 , G x = — 1 , G 3 = (— 4) B l5 G 3 =0, G 4 = (— 4) 3 B 3 , G 5 = 0, 
G 6 = (— 4) 3 B 3 , . . . then, using /3 K in the same sense as at p. 289, we shall find the 
scale of relation between the B’s (Bernoulli’s Numbers) is given by the equation 
ac = i 
X (— Yfri. G i_ K = 0, provided i is odd. 
K = 0 
On striking out the i which intervenes between and G;_„ so as to make the 
former operate on the latter, the equation becomes that given at p. 289 for the E’s, 
the sharpened numbers of Hamilton. —J. J. S. 
